An Improvement-Path Framework and an Exact Algorithm for Single-Machine Scheduling with Release Times

This paper studies the non-preemptive single-machine scheduling problem with heterogeneous release times and processing times, with the objective of minimizing total waiting time. The problem is known to be NP-hard. By modeling machine idle time as negative waiting time, we establish a unified waiting-time formulation that compresses the original four-dimensional description into a two-dimensional representation depending only on processing times and waiting times, thereby substantially simplifying the structural analysis of the problem. Based on this representation, ideal directions are defined from the structure induced by an optimal solution, and it is shown that every non-optimal schedule admits an ideal improvement path. It is further proved that improvable ideal directions require no coordination and can therefore be treated as independent improvement units. The theoretical analysis also shows that the realizability of every improvable ideal direction can be determined in finitely many steps, thereby laying the foundation for an overall solution framework. For each improvable ideal direction, by analyzing the propagation of the decrease flow and the increase flow under local adjustments, we prove that queue discontinuity is the unique structural obstacle to the realization of ideal improvement paths. A complete characterization of the resulting local optima is then established, and corresponding repair operators are designed. On this basis, an iterative repair algorithm under the improvement-path framework is developed. It is proved to terminate in finitely many steps, return a globally optimal schedule, and admit an explicit upper bound on its time complexity. This work provides a new analytical perspective and solution framework for this class of NP-hard scheduling problems.